Tsim, Tsev kawm ntawv qib thiab universities
Yuav ua li cas yog ib tug floating taw tes tooj?
Kev nthuav qhia ntawm tiag tiag (los yog tiag tiag) cov zauv, qhov uas lawv yuav muab raws li ib tug mantissa thiab exponent yog floating taw tes tus xov tooj (tej zaum point, raws li yog ib txoj kev cai nyob rau hauv lub English-hais lus lub teb chaws). Dua li no, tus xov tooj yog muab nrog ib taag txheeb ze raug thiab hloov tsis. Sawv cev uas yog siv ntau, pom zoo standard IEEE 754. lej haujlwm uas siv floating-point tus xov tooj yog siv nyob rau hauv koj laij tshuab - ob hardware thiab software.
Point los yog tsis nco qab
Cov ncauj lus kom ntxaws daim ntawv teev cov zauv separator qhia cov neeg kawm lus Askiv hais lus lub teb chaws thiab anglofitsirovannye, qhov twg cov ntaub ntawv ntawm tus xov tooj sib cais los ntawm ib tug fractional ib feem ntawm tag nrho cov point, vim hais tias cov terminology ntawm cov teb chaws tau saws lub npe floating taw tes - "floating point". Nyob rau hauv lub Lavxias teb sab Federation, lub fractional ib feem ntawm tag nrho cov kev lig kev cai, sib cais los ntawm ib tug tsis nco qab, li ntawd nws nruab nrab yog cov tib lub tswvyim tau dabneeg pom tau hais tias lub sij hawm "floating point". Txawm li cas los, hnub no nyob rau hauv cov kev cov ntaub ntawv thiab nyob rau hauv Lavxias teb sab ntaub ntawv nws yog tau tso cai ob xaiv.
Lub sij hawm "floating point" originated los ntawm lub fact tias ib tug positional tooj sawv cev yog ib tug tsis nco qab (qub decimal los yog binary - ib tug computer) uas yuav haum txhua qhov chaw ntawm cov kab zauv. Qhov no feature yog kom stipulate nws nyias. Qhov no txhais tau tias cov sawv cev ntawm floating taw tes tus xov tooj yuav tsum tau ntshai raws li ib tug computer siv ntawm exponential cim. Qhov kom zoo dua ntawm siv xws li ib tug sawv cev ntawm ib tug sawv cev hom tsau-point thiab integer xov tooj uas ntsuas ntawm qhov tseem ceeb hlob ho thaum uas txheeb ze raug tseem unchanged.
Piv txwv li
Yog hais tias lub comma nyob rau hauv lub xov tooj ntawm tsau, ces hlawv nws tsuas yog ib hom ntawv. Piv txwv li, muab ib tug me ntsis ntawm ib tug rau hauv xov tooj thiab ob tug lej nyob rau hauv lub fractional ib feem. Qhov no yuav ua tau xwb nyob rau hauv no txoj kev: 123456,78. Cov hom ntawv ntawm floating taw tes tus xov tooj muab tag nrho tau rau kev qhia. Piv txwv li, muab lub tib yim xaus-npawb tsab. Ntaubntawv povthawj siv kev xaiv tej zaum yuav muaj yog hais tias tus programmer tsis ua ib tug ob tug skimp luag hauj lwm ntxiv teb, nyob qhov twg nws yuav sau cov exponents uas yog feem ntau 10, thiab los ntawm 0 mus txog 16, thiab ntswg thaum tag nrho cov yuav muaj kaum 8 + 2.
Ib txhia embodiments ntawm cov ntaubntawv povthawj siv, uas tso cai rau koj mus format xov tooj nrog floating point: 12345678000000000000; 0,0000012345678; 123,45678; 1.2345678 thiab thiaj li nyob. Nyob rau hauv no hom, muaj txawm tias ib tug unit ntawm kev ntsuas ntawm kev ceev! Tiam sis, tus ua tau zoo ntawm ib tug computer system uas sau ceev uas lub computer ua hauj lwm qhov chaw uas muaj cov sawv cev ntawm floating taw tes tus xov tooj. Qhov no ua tau zoo yog ntsuas nyob rau hauv cov nqe lus ntawm flops (floating-point ua hauj lwm rau ib tug thib ob, uas translates rau tus xov tooj ntawm muas ib ob nrog ib tug floating taw tes). Qhov no yog lub hauv paus nyob rau hauv cov kev ntsuas computer system ceev.
qauv
Cov ntaub ntawv xov tooj nyob rau hauv lub floating taw tes hom ntawv yog tsim nyog raws li nram no, tsom kwm ib theem zuj zus ntawm lub yuav tsum tau qhov chaw, vim hais tias qhov no cov ntaub ntawv yog exponential, uas qhia tau hais tias cov zauv tiag tiag raws li ib tug mantissa thiab kev txiav txim. Nws yog tsim nyog los sawv cev rau ib yam nkaus thiab loj loj thiab me me heev cov xov tooj, lawv yog yooj yim npaum li mus nyeem. Yuav tsum tau qhov chaw: lub kaw xov tooj (N), lub mantissa (M), qhov kev txiav txim ntawm tus kos npe rau (p) thiab qhov kev txiav txim (n). Lub ob nta ntawm lub kos npe rau. Li no, N = M. n p. Yog li ntawd sau floating-point xov tooj. Piv txwv yuav varied.
1. Nws yog tsim nyog los sau cov xov tooj ntawm ib tug lab, thiaj li tsis tau poob rau hauv lub zeros. 1000000 - nws yog ib tug ib txwm cov ntaubntawv povthawj siv, xam. A computer yog raws li nram no: 1.0. Lub kaum hli ntuj 6. Hais tias yog, kaum mus rau lub thib hwj chim - peb tej yam tshwm sim, uas haum nyob rau hauv raws li ntau raws li rau zeros. Yog li no tshwm sim sawv cev ntawm tus xov tooj ntawm taag thiab floating taw tes qhov twg tam sim ntawd yuav ntes txawv nyob rau hauv txawv.
2. Thiab xws li ib tug nyuaj tooj yog 1,435,000,000 (ib billion plaub puas thiab peb caug-tsib txhiab) kuj yuav sau ib tsab: 1.435. Cuaj hlis 10, xwb. Yog li ntawd nws yog nrog ib tug rho tawm kos npe rau yuav sau tej xov tooj. Tus ntawd yog nws, thiab txawv los ntawm txhua lwm yam nrog rau cov xov tooj ntawm taag thiab floating taw tes.
Tab sis nws yog ntau tshaj ntawm yuav ua li cas rau tsis muaj? Yog, yooj yim heev.
3. Piv txwv li, raws li ib tug millionth cim? = 0.000001 1.0. 10 -6. Heev thawm ntawv thiab sau ntawv xov tooj, thiab nyeem nws.
4. Ib tug ntau nyuab? Tsib puas thiab plaub caug-thib rau lub billionth: 0.000000546 = 546. 10 -9. Ntawm no. Cov kab kev los ntawm floating taw tes yog dav heev.
zoo
Daim ntawv xov tooj tej zaum yuav ua tau li qub los yog normalized. Qhov - yeej ib txwm hwm tus precision ntawm floating taw tes tus xov tooj. Nws yuav tsum tau muab sau tseg tias mantissa nyob rau hauv daim ntawv no, tsis tau noj mus rau hauv tus account lub kos npe rau, yog ib nrab ntawm lub luv 0 1, ces 0 ⩽ ib tug <1. Tsis nyob rau hauv lub qub daim ntawv ntawm tus xov tooj ntawm loses nws cov neeg. Lub disadvantage ntawm lub cev daim ntawv no yog hais tias muaj ntau tus xov tooj yuav tsum muab sau nyob rau hauv ntau txoj kev, uas yog tsis meej xwm. QAUV txawv cov ntaub ntawv ntawm tib tooj: 0 = 0.0001, 000001. 10 2 = 0.00001. Lub ib hlis ntuj 10 = 0.0001. 10 0 = 0.001. 10 -1 = 0.01. 10 -2, thiab thiaj li yuav ua tau ntau npaum li cas. Uas yog vim li cas lub computer siv ib tug txawv normalized cim, qhov chaw uas lub mantissa decimal assumes tus nqi ntawm cov chav nyob (inclusive), thiab yog li rau kaum (tsis muaj), thiab nyob rau hauv tib txoj kev lub mantissa binary tooj muaj ib tug nqi ntawm ib tug (inclusive) rau ob (tsis inclusive).
Yog li ntawd, 1 ⩽ a <10. Qhov no - binary xov tooj nrog floating taw tes, thiab daim ntawv no ntawm cov ntaubntawv povthawj siv tej xov tooj (tsuas yog zero) captures ib tug tshwj xeeb txoj kev. Tab sis kuj muaj yog ib tug drawback - lub cev tsis xav txog tej yam no zoo pes tsawg. Yog li ntawd Informatics muab rau cov kev siv tshwj xeeb xov tooj 0 kos npe rau (me ntsis). Cov integer ib feem ntawm (MSB) ntawm lub mantissa nyob rau hauv lub binary tooj tsuas yog pes tsawg nyob rau hauv ib tug normalized daim ntawv no yog sib npaug zos rau 1 (implicit unit). Qhov no cov ntaub ntawv no yog siv txuj IEEE 754. Cov positional tooj system, nyob rau lub hauv paus yog ntau tshaj ob (ternary, quaternary thiab lwm yam systems), cov cuab yeej no yog tsis yuav.
reals
Cov zauv tiag tiag nrog floating taw tes thiab yog feem ntau yog li no nws tsis yog tsuas yog ib, tab sis ib tug heev yooj yim txoj kev uas yuav sawv cev rau ib tug tiag tiag tooj, raws li nws, ib tug nruab nrab ntawm cov ntau yam ntawm qhov tseem ceeb thiab yog cov neeg. Qhov no yog analogous mus exponential cim, tsuas yog ua rau lub computer. Floating-point tooj - ib tug txheej ntawm ib tug neeg me me yog muab faib mus rau hauv ib tug kos npe rau (kos npe rau), kev txiav txim (exponent) thiab mantissa (ntsuas). Feem ntau hom ntawv yog ib tug IEEE 754 floating-point tooj raws li ib tug txheej ntawm cov khoom uas encode ib feem ntawm nws cov mantissa, lwm tus ib feem - cov neeg kawm ntawv thiab ib tug me ntsis qhia lub kos npe rau ntawm tus xov tooj: pes tsawg - yog hais tias nws yog zoo, lub tsev - yog hais tias lub xov tooj yog tsis zoo. Tag nrho txoj kev yog sau cia los ntawm ib tug xov tooj (code-ua haujlwm), thiab lub mantissa - nyob rau hauv ib tug normalized daim ntawv no, nws fractional ib feem - nyob rau hauv lub binary system.
Txhua kos npe rau - yog ib tug hluas me ntsis uas hais tias tus kos npe rau tag nrho cov floating-point xov tooj. Mantissa thiab kev txiav txim - yog zauv, lawv, nrog rau tus kos npe rau thiab ua tus sawv cev ntawm floating taw tes tus xov tooj. Tus txheej txheem yuav tsum tau hu ua ib tug exponential los yog exponent. Tsis yog txhua txhua cov zauv tiag tiag yuav tsum sawv cev nyob rau hauv ib tug computer nyob rau hauv lawv cov caij nyoog lub ntsiab lus, lwm tus uas hais approximate qhov tseem ceeb. Ib tug ntau yooj yim xaiv - xa ib tug tiag tiag tooj nrog ib taag taw tes, nyob qhov twg tiag tiag thiab tag nrho ib feem yuav tsum khaws tseg cia cais. Feem ntau cov yuav, yog li ntawd lub integer ib feem yog ib txwm allotted X me me, thiab ib tug fractional - Y khoom. Tab sis lub architecture ntawm processors yog tsis paub txog xws li ib tug txoj kev, tab sis vim hais tias nyiam yog muab rau tus xov tooj ntawm floating taw tes.
Tsis tas li ntawd
Tsis tas li ntawd ntawm floating taw tes tus xov tooj yog heev yooj yim. Nyob rau hauv kev twb kev txuas nrog cov IEEE 754 standard ib precision tooj nws muaj ib tug loj tus naj npawb ntawm cov khoom, li ntawd nws yog zoo dua rau koj tsiv nyob rau cov piv txwv, muaj ib tug zoo dua lub tswv yim coj qhov tsawg tshaj plaws floating-point tooj. Piv txwv li, lub ob tug zauv - X thiab Y.
| nce mus nce los | cim | exponent | mantissa |
| X | 0 | 1001 | 110 |
| Y | 0 | 0111 | 000 |
Cov kauj ruam yog raws li nram no:
a) Cov zauv yuav tsum tau sawv cev nyob rau hauv normalized daim ntawv. Nws yog kom meej meej ib tug thaum ib tug. X = 1,110. 2 2, thiab Y = 1,000. 2 0.
b) Mus hauv tus txheej txheem ntawm muaj pes tsawg leeg yuav tsuas equalize lub tso rau neeg pom, tab sis nws yuav tsum tau mus sau dua tshiab tus nqi ntawm Y. Nws yuav sib haum mus rau tus nqi ntawm cov normalized xov tooj, txawm hais tias nyob rau hauv qhov tseeb - unnormalizes.
Xam qhov txawv nruab nrab ntawm tus exponents ntawm degree 2 - 0 = 2. Tam sim no tsiv mus nyob rau mantissa pab kom txawm peem rau cov kev hloov, uas yog, ntxiv 2 mus rau lub Performance index ntawm lub thib ob lub sij hawm, yog li tsiv ib comma thaum units ntawm ob lub ntsiab lus mus rau sab laug. 0,0100 yog tau. Lub ob hlis ntuj 2. Qhov no yuav ua tau qhov sib npaug ntawm yav tas los tus nqi Y, ces muaj yog ib tug twb tau Y '.
c) Tam sim no koj yuav tsum ntxiv txog cov xov tooj ntawm mantissa X thiab Y. tom
1,110 + 0,01 = 10,0
Exhibitor tseem yog sawv cev los ntawm lub X parameter, uas yog sib npaug zos rau 2.
g) Tus nqi uas tau txais nyob rau hauv lub yav dhau los kauj ruam, txav lub normalization unit, ces koj yuav tau hloov lub exponent sum thiab hais dua. 10.0 nrog ob tug me me mus rau sab laug ntawm tus zauv point, tus xov tooj yog tam sim no tsim nyog rau normalize, piv txwv li, txav mus rau tsis nco qab mus rau sab laug los ntawm ib tug taw tes, thiab exponent, ntsig txog, nce los ntawm 1. Nws puv 1,000. Lub peb hlis ntuj 2.
e) Nws yog lub sij hawm mus hloov ib tug floating taw tes tooj nyob rau hauv ib-byte system.
| sum | cim | exponent | mantissa |
| X + Y | 0 | 1010 | 000 |
xaus
Raws li koj tau pom, ntxiv cov zauv tsis nyuaj, dab tsi uas floats comma. Tshwj tsis yog tias, ntawm chav kawm, tshwj tsis yog rau koj nqa lub xov tooj ntawm sab exponent cov ntau (nyob rau hauv lub saum toj no piv txwv li, nws yog tus Y kom X), raws li zoo raws li lub restoration txog tus txheej xwm quo, piv txwv li lub qhov teeb meem ntawm nyiaj - tsiv mus nyob rau cov zauv point mus rau sab laug ntawm lub mantissa. Thaum lub tas li ntawd twb tau thov, nws yog heev sai tau thiab tseem yog ib qho teeb meem - perenormirovanie thiab truncation ntsis yog tias lawv muaj pes tsawg tsis phim lub xov tooj los sawv cev rau nws.
npaug
Binary system muaj ob txoj kev los ntawm cov uas multiplies cov floating-point xov tooj. Qhov no ua hauj lwm tej zaum yuav tsum tau ua los ntawm kev ua lej khu, uas yuav pib nrog lub kawg tseem ceeb khoom thiab cov uas yuav pib nrog lub siab thiaj me me nyob rau hauv lub d. Ob tus neeg mob muaj ib tug xov tooj ntawm haujlwm sequentially stacking ib nrab khoom. Cov hauj lwm yog tswj los ntawm qhov sib ntxiv ntawm d khoom. Yog li ntawd, yog ib tug ntawm cov khoom ntawm cov d yog ib chav tsev, cov sum ntawm ib nrab khoom ntawm lub multiplicand hlob nrog ib tug coj ua haujlwm. Yog hais tias ib tug lej ntawm lub d crept pes tsawg, thaum lub sij hawm multiplicand yog tsis ntxiv.
Yog hais tias npaug yog ua cia li ob lub xov tooj, cov khoom ntawm tus xov tooj nyob rau hauv nws cov nyiaj yuav tsis pub tshaj tus naj npawb ntawm cov lej tom qab muaj nyob rau hauv cov yam tseem ceeb, ntau tshaj ob zaug, thiab rau coob nws yog heev, heev npaum li cas. Yog hais tias khoo los ntawm ib co xov tooj, cov khoom txaus ntshai tsis haum rau qhov kev tshuaj ntsuam. Vim hais tias cov xov tooj ntawm cov khoom ntawm tej cov tshuab yog heev finite, thiab nws forces rau kaw ib tug tshaj plaws ntawm ob npaug ntawm cov adders tug lej tom qab. Thiab yog hais tias lub xov tooj ntawm qhov chaw yog tas, nyob rau hauv cov khoom yuav inevitably paub tsis. Yog hais tias tus nqi ntawm cov le caag yog loj, qhov yuam kev ntawm sib tshooj, thiab raws li ib tug tshwm sim zoo heev tsub kom lub zuag qhia tag nrho cov neeg. Ntawm no, tsuas yog txoj kev - mus nyob ib puag ncig lub npaug tau, ces qhov yuam kev tej hauj lwm twb alternating. Thaum ib tug npaug lub lag luam, nws yuav tau mus tshaj cov kab sib chaws ntawm tug lej tom qab, tiam sis tsuas yog tus yau, vim hais tias muaj yog ib tug tsis pub yuam rau tus xov tooj uas yog sawv cev nyob rau hauv daim ntawv ntawm cov tsau-point.
ib co lus qhia
Zoo pib los ntawm thaum pib. Feem ntau txoj kev los sawv cev rau tus xov tooj - kab zauv raws li ib tug integer, qhov twg tsis nco qab yog rub nyob rau hauv lub heev kawg. Qhov no txoj hlua yuav ua tau tej ntev, tab sis ib tug tsis nco qab sawv nyob rau hauv txoj cai qhov chaw muab tso rau nws, separating lub integer los ntawm cov fractional ib feem ntawm nws. Cov hom ntawv ntawm kev nthuav qhia ntawm qhov chaw ruaj-point system tas txo nws hwj tej yam kev mob nyob rau hauv lub qhov chaw ntawm tus zauv point. Kev cim siv ib tug txheej txheem normalized saib ntawm cov sawv cev ntawm cov zauv. Nws aqn {\ displaystyle aq ^ {n }} aq n. Ntawm no ib tug {\ displaystyle ib tug} ib tug, thiab nws yog hu ua tus mantissa lace. Cia li hais txog nws twb tau hais tias 0 ⩽ ib tug
Floating-point tooj yog sau zoo heev rau tag nrho cov meej tus qauv nkag xov tooj, tsuas yog tus exponent thiab mantissa yog kaw nyias. Xeem los tib yam thiab nyob rau hauv ib tug normalized hom - tsau taw tes, uas yog dai kom zoo nkauj nrog rau cov thawj tseem ceeb lej. Cia li floating taw tes yog siv feem ntau nyob rau hauv lub computer, uas yog, nyob rau hauv lub tshuab hluav taws xob sawv cev ntawm qhov chaw uas tus system yog tsis zauv thiab binary, qhov twg txawm mantissa Denormalize lawm point - tam sim no nws yog ua ntej thawj tug lej, ces ua ntej, tsis tau tom qab nws, qhov twg integer ib feem nyob rau hauv tus, yuav ua tsis tau. Piv txwv li, peb tus kheej decimal system yuav muab nws cuaj binary system rau ib ntus siv. Thiab hais tias yuav sau thiab nws cov mantissa floating-point zoo li no: +1001000 ... 0, thiab nws thiab cov index 0 ... 0100. Tab sis tus zauv system tsis ua xws complex suav, uas tej zaum yuav nyob rau hauv binary, siv rau hauv daim ntawv ntawm floating taw tes.
ntev xam
Nyob rau hauv hluav taws xob computers tau ua-nyob rau hauv software pob, qhov twg faib rau cov mantissa thiab exponent ntawm tus nqi ntawm lub cim xeeb teev software, tsuas los ntawm lub cim xeeb loj ntawm lub computer. Nws zoo li ib tug ntev lub xam, uas yog, yooj yim rau tus xov tooj uas ua computer. Nws yog tag nrho cov tib - sib rho thiab tsis tas li ntawd, division thiab lej khu, elementary zog thiab kev tsim kho ntawm lub hauv paus. Tab sis lub xov tooj ntawm txawv heev, lawv muaj peev xwm yog ho ntau dua tshaj qhov ntev ntawm lub tshuab lo lus. Cov kev siv ntawm cov ua hauj lwm tsis yog los ntawm hardware thiab software, tiam sis nws yog dav siv yooj yim kho vajtse ua hauj lwm nrog me npaum li cas tus xov tooj ntawm kev txiav txim. Muaj ntau thiab xam, qhov twg tus xov tooj ntev tsuas txwv los ntawm lub cim xeeb muaj peev xwm - arbitrary precision xam. Ib tug ntev xam yog siv nyob rau hauv ntau lub teb.
1. Yuav kom ua ke lub code (processors, microcontrollers nrog tsawg me ntsis tob - 10-ntsis sau thiab yim-ntsis lo lus ntev, nws yog tsis txaus rau lis cov ntaub ntawv los ntawm lub Analog-rau-digital (analog-rau-digital converter), thiab yog li ntawd tsis tau ua tsis muaj ib tug ntev xam.
2. Nws yog tseem yog ib tug ntev xam yog siv rau cryptography, qhov twg nws yog tsim nyog los xyuas kom meej qhov tseeb ntawm cov kev tshwm sim ntawm exponentiation los yog sib npaug rau 10.309. Integer arithmetic yog siv modulo m - ib tug loj tus naj npawb loj, thiab tsis tas yoojyim.
3. Kev software rau cov neeg ua haujlwm thiab cov neeg ua lej kuj tsis tuaj yeem ua tsis tau ntev ntev, vim hais tias tsuas yog ua li no thiaj li paub tseeb cov lus teb rau ntawm daim ntawv - siv lub tshuab computer, muab qhov tseeb ntawm cov lej. Floating taw tes lawv tuaj yeem rub kom ntev li ntev tau. Tab sis engineering calculations thiab cov hauj lwm ntawm cov kws tshawb fawb tsis tshua muaj kev cuam tshuam ntawm software suav, vim nws yog qhov nyuaj heev los ua cov lus tsis ua yuam kev. Feem ntau lawv loj dua cov qhab nia ntawm kev sib tw.
Sib ntaus yuam kev
Hauv kev khiav lag luam nrog cov zauv uas lub comma floating, nws yog qhov nyuaj heev los soj ntsuam qhov yuam kev ntawm kev soj ntsuam. Li ntawd, nyob deb, muaj tsis tau tsim ib lub zauv zeem txaus siab txhua yam uas yuav pab daws qhov teeb meem no. Tab sis cov teeb meem nrog cov zauv cim yog cov yooj yim rau kev ntsuam xyuas. Tau tshem ntawm inaccuracies lies nyob rau saum npoo - tsuas yog siv cov zauv xwb nrog ib lub voos xwm txheej tas. Piv txwv li, cov kev pabcuam nyiaj txiag tau ua ntawm txoj cai no. Txawm li cas los, nws yog yooj yim: qhov yuav tsum muaj pes tsawg tus zauv tom qab qhov zauv zuag qhia paub ua ntej.
Lwm cov kev siv tsis muaj peev xwm txwv rau qhov no, vim hais tias nws tsis muaj peev xwm ua hauj lwm nrog cov zauv me me los yog ntau heev. Yog li, thaum ua hauj lwm, nws yog ib txwm coj mus rau hauv tus account tias inaccuracies yog ua tau, thiab yog li ntawd, thaum muab tawm qhov kev tshwm sim, nws yog ib qhov tsim nyog yuav tau muab tso tawm. Thiab, kev hloov tsis zoo no feem ntau ua tsis txaus, thiab yog li ntawd muab faib tshwj xeeb. Kev sib piv cov lag luam yog qhov txaus ntshai heev rau hauv no. Ntawm no, txawm tias yuav ntsuam xyuas qhov loj me ntawm yav tom ntej qhov teeb meem yog qhov nyuaj kawg.
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